It came as an immediate split emerged between Mr Corbyn and Tom Watson, the new deputy leader, over renewal of the Trident nuclear deterrent. Mr Watson also made clear that he would oppose Mr Corbyn’s positions on the European Union and Nato, in a sign that the new leadership may be unable to agree on key policy positions in the weeks ahead.

Mr Corbyn refused to be interviewed on the BBC’s Andrew Marr programme, instead choosing to appear at a small rally to discuss mental health in his north London constituency.

He spent much of the day in Parliament, trying to form a shadow cabinet following the resignation of more than 10 shadow ministers in the hours after he was declared Labour leader on Saturday. He was forced to offer potential shadow cabinet members free votes on issues such as Trident because so many moderate MPs oppose his views on the military and the EU.

In one of his first acts, the new Labour leader sacked Ivan Lewis, one of his critics, as shadow Northern Ireland secretary. Mary Creagh, the europhile former shadow international development secretary, announced she was returning to the back benches.

However, Mr Corbyn was eventually able to appoint a number of MPs to his shadow cabinet. Andy Burnham, who Mr Corbyn defeated in the leadership contest, was named shadow home secretary. Hilary Benn remains as shadow foreign secretary, while Heidi Alexander was given the shadow health brief and Lord Falconer, a friend of Tony Blair, becomes shadow justice secretary.

Yvette Cooper refused to take a front bench role but will lead Labour’s “refugee task force”.

Mr Corbyn will face his first test as Labour leader today, when the Tories put forward their Trade Union Bill, which is designed to dramatically reduce the influence of unions and their ability to call strikes.

As part of the reforms, the Conservatives are planning a four-month time limit on ballots so that mandates for industrial action are always recent. Currently, unions are able to hold strikes years after balloting members.

At the TUC annual conference in Brighton, union leaders lined up to call for strikes and “occupations” following Mr Corbyn’s victory, as they vowed to oppose the new Tory legislation.

Len McCluskey, the Unite leader

Mark Serwotka, leader of the PCS union, said: “You have to pinch yourselves that a Labour leader is saying things that all of us agree with.

“If we are going to see any of those policies realised, we will not get that just through what Jeremy Corbyn and the Labour Party do in Parliament. If Jeremy Corbyn wants to win on those policies, he absolutely needs a mass vibrant movement in the country… He needs the six-and-a-half million trade union members to ensure that we have that vibrant campaign through strikes, demonstrations, local campaigns, occupations and everything else.”

He added: “We have the ability to stop austerity in its tracks, to topple this government and to ensure we get a fairer society.”

Len McCluskey, the Unite leader, personally congratulated Mr McDonnell, Mr Corbyn’s chief of staff, and said that the Tories “will never ever, ever defeat a united working class, a united trade union movement, a united Labour movement”.

Rob Williams, of the National Shop Stewards Network, said his members would “take down” Mr Cameron. He said: “The victory yesterday by Jeremy Corbyn has changed everything. The vote we saw yesterday was a political revolution. We must build a mass movement against austerity and the anti-union laws.

“The message must be simple – 'Cameron: we are going to take you down. Your anti-union Bill and your cuts, you’re going down because we are mobilising against you’.

“If this goes into law, we want mass coordinated strike action.”

Ronnie Draper, leader of the bakers’ union, said that he and “thousands” of other union members were prepared to go to jail for coordinating illegal strikes if the new Tory laws are passed.

He said: “I don’t want to go to jail comrades – but I tell you, I will. And there will be thousands of others.”

The new Labour leader is also facing accusations of hypocrisy from his supporters after his aides said he was now accepting a post on the Queen’s Privy Council.

It had previously been suggested that Mr Corbyn, a staunch republican, would refuse the role, which automatically comes with being the Opposition leader.

In a damaging blow for Mr Corbyn, Mr Mills, who gave £1.65 million to the party under Ed Miliband, said that Labour now risks becoming a “protest” party.